Explorations and Reasoning in the Dynamic Geometry Environment

This article is to describe how the author took full advantage of the exploration feature of the Geometer’s Sketchpad, a dynamic geometry software package, to help the preservice secondary school mathematics teachers develop their good learning habits such as making and verifying conjectures, as well as their mathematical reasoning and proof abilities. Three examples are presented to show the role of students’ extensive GSP explorations – helping them discover important, interesting mathematical facts/ideas, which in turn became an impetus for generating proofs; and providing insights for them to come up with proof ideas. Geometry is a weak spot in school mathematics education. Research (see [2] and [8]) has indicated that students entering high school have very little knowledge or experience of geometric properties and relationships; most are operating at the visual level of geometric thought. These students can do little more than recognize different geometric shapes. They do not realize that a square is also a rectangle and a rhombus, or that all three are also parallelograms. Many do not realize that a square must have four right angles or that all four sides are congruent. Most have never heard of a “line of symmetry”, let alone understand why a square has four lines of symmetry whereas a non-square rectangle has only two (see [5]). The problems that students have with perceiving a need for proof are well known to all high-school teachers and have been identified without exception in all education research as a major problem in the teaching of proof (see [1]). To improve this situation, the effective use of technology seems to be a solution. Peressini & Knuth (see [6]) have formulated five primary ways in which technology is currently being used as a pedagogical tool in mathematics classrooms. After listing the four other roles technology plays in the mathematics classroom – as a management tool, as a communication role, as an evaluation tool, and as a motivational tool, they indicate, The fifth role, and perhaps the most important role from the perspective of school mathematics reform, is to harness technology in ways that help students better understand mathematical algorithms, procedures, concepts, and problem-solving situations. In this capacity, as a cognitive role, technology offers a unique means for supporting students’ exploration of, and engagement with, mathematics. It affords “new ways of representing complex concepts, and makes available new means by which students [or teachers] can manipulate abstract entities in a ‘hands-on’ way” (p.280). Dynamic Geometry is active, exploratory geometry carried out with interactive computer software. As Goldenberg & Cuoco (see [3]) point out, the term dynamic geometry has quickly entered the literature as a generic term because of its aptness at characterizing the feature that distinguished this geometry from other geometry: the continuous real-time transformation often called “dragging.” This feature allows users, after a construction is made, to move certain elements of a drawing freely and to observe other elements respond dynamically to the altered conditions. As these elements are moved smoothly over the continuous domain in which they exist, the software maintains all relationships that were specified as essential constraints of the original construction, and all relationships that are mathematical consequences of these. Hence the software allows a focus on the important geometrical idea of invariance. The nature of the dynamic geometry software also makes it conducive to collaborative problem solving among small or even large groups of students. Several students gathered around a single computer are easily caught up in the conjecturing process as they watch the changes taking place on the computer screen and are quick to offer suggestions for further experimentation (see [5]). This article is to describe how I took full advantage of the exploration feature of the Geometer’s Sketchpad (see [4]), a dynamic geometry software package, to help the preservice secondary school mathematics teachers develop their good learning habits such as making and verifying conjectures, as well as their mathematical reasoning and proof abilities. My approach was to present problems to students not in a format like “Prove the following fact (or statement)” but in such a way that the students needed to construct the problem situation (generally a figurative representation of the given problem), observe the situation and manipulate the components of the situation (such as dragging a point or a line and doing some measurements), and answer questions such as “What do you notice?” and “Can you prove your findings?” My practice has shown that this inquiry-based, problem-solving approach can allow students to “manipulate abstract entities in a ‘hands-on’ way”, stimulate their curiosity of finding “why”, provide them with more flexibility to come up with proof insights, and help them achieve better, conceptual understanding. 1. Discoveries – Curiosity – Proof Insights The first example is the following problem describing an old pirate parchment: The island where I buried my treasure contains a single palm tree. Find the tree. From the palm tree, walk directly to the Eagle-shaped rock. Count your paces as you walk. Turn a quarter circle to the right and go the same number of paces. When you reach the end, put a stick in the ground. Return to the palm tree and walk to the Owl-shaped rock, again counting your paces. Turn a quarter circle to the left and go the same number of paces. Put another stick in the ground. Connect the sticks with a rope and dig beneath its midpoint to find the treasure. If the two rocks remain but the palm tree has long since died, can the treasure still be unearthed? I asked the students (the preservice teachers, the same hereafter) to create a Geometer’s Sketchpad (GSP) representation of the problem situation. Following the instructions written on the pirate parchment, the students quickly constructed a situation similar to the one shown in Figure 1: m EO = 12.39 cm m TO = 8.76 cm m ET = 8.76 cm m∠ETO = 90.00° Given: PE ⊥ ES1 and PO ⊥ OS2 PE ≅ ES1 and PO ≅ OS2 T is midpoint of S1S2 E and O are arbitrary but fixed. Find: If P is removed, can we still find T? P=Palm Tree E=Eagle-shaped Rock O=Owl-shaped Rock S1=Stick S2=Stick T=Treasure